Efficent method of identifying non-solution or non-optimal regions of the domain of a function

ABSTRACT

A method of identifying one or more regions of the domain of a function that do not contain solutions is described along with a related subdivision method. These methods may be employed in the context of branch and bound methods that use interval analysis to search for solutions of functions. The one or more regions of the function domain that do not contain solutions are identified using a cropping formula derived from one or more components (low order and high order) of a Taylor Form inclusion function. A Corner Taylor Form inclusion function is also described which might be used to identify the output range of a function.

[0001] This application claims the benefit of U.S. ProvisionalApplication No. 60/234,829, filed Sep. 22, 2000, and U.S. ProvisionalApplication No. 60/233,545, filed Sep. 19, 2000, which are both herebyfully incorporated by reference herein as though set forth in full.

FIELD OF THE INVENTION

[0002] This application relates generally to the field of searchprocesses for locating solutions (defined to include optimized solutions(subject to constraints), roots, minima, or maxima) of functions, and,more specifically, to methods of identifying solution containing regionsby elimination of non-solution containing regions of the domain of thefunctions.

RELATED ART

[0003] Branch and bound search methods are methods for searching forsolutions to functions, such as complex non-linear functions, that defytraditional closed-form techniques for finding solutions. Conventionalbranch and bound solution searching methods find solutions to thefinctions by iterating over potential solution-containing regions of theinput variables to the functions. During an iteration, if a region isdetermined to define a solution, it is added to a collection ofsolutions; if it is guaranteed not to contain a solution, it is removedfrom further consideration; and if it may contain a solution, it issubdivided, and additional iterations of the method performed over thesubdivided regions. The methods continue until all potentialsolution-containing regions have been considered.

[0004] A problem with conventional branch and bound solution searchingmethods is they do not rapidly and efficiently converge tosolution-defining regions of the functions. Consequently, many attemptshave been made to increase the speed of convergence of the branch andbound methods. The convergence speed of these methods depends on howefficiently one can detect if a region is guaranteed not to contain anysolutions.

[0005] A widely used technique in these searching methods is the use ofinterval analysis to determine if a region does or does not contain asolution. Interval analysis is a branch of applied mathematics thatoriginated with the work of R. E. Moore in the 1960s. See, e.g., Moore,R. E., Interval Analysis, Prentice Hall, 1966. In interval analysis,functions known as inclusion functions are used to determine the rangeof the function output which results from input variables which areimprecise or expressed in terms of intervals, ranges or regions. Theoutput range is then examined to determine if it contains or does notcontain a solution (minimum or maximum) of the function. If not, thecorresponding intervals, ranges or regions of the input variables areexcluded from further consideration. However, intervals, ranges orregions which include possible solutions may then be subdivided,typically using a binary subdivision technique, and additionaliterations of the searching algorithm performed on the subdivided areas.The iterations may continue until a solution is located or the search isexhausted.

[0006] Referring to FIG. 1, a flowchart of a conventional branch andbound search method employing interval analysis is illustrated.

[0007] As illustrated, the method begins in step 102, in which a queryis made whether an input queue of potential solution-containing regionsis exhausted, or whether there are still one or more potentialsolution-containing regions on the input queue. If the input queue isexhausted, the method terminates. On the other hand, if the input queuecontains one or potential solution-containing regions, step 104 isperformed. In step 104, a potential solution-containing region isselected from the input queue.

[0008] Step 106 is then performed. In step 106, a function output rangecorresponding to a potential solution-containing region is determinedusing an inclusion function.

[0009] The function output range is then analyzed to determine if thecorresponding potential solution-containing region contains, defines, ordoes not contain a solution of the function. If the potentialsolution-containing region contains a solution of the function, step 110is performed. If the potential solution-containing region defines asolution of the function, step 112 is performed. If the potentialsolution-containing region does not contain a solution of the function,step 114 is performed.

[0010] In step 110, the solution-containing region is subdivided, andthe subdivided portions placed on the input queue. Then, anotheriteration of the method is performed by looping back to step 102.

[0011] In step 112, the potential solution region is classified as asolution by placing it on a solution queue, and removing it from theinput queue. The method then loops back to step 102 to perform anotheriteration.

[0012] In step 114, the potential solution region is discarded from theinput queue, and the method loops back to step 102 for anotheriteration.

[0013] Unfortunately, to date, interval analysis has been applied tocertain applications such as computer graphics with mixed results. Inparticular, the searching methods are still too slow to be viable insuch applications because of the real-time constraints that are inherentin these applications. The time and memory requirements of these methodsare exponential in the number of digits of precision required as well asin the number of free variables of the functions considered.

SUMMARY

[0014] The invention provides a method for efficiently identifyingnon-solution-containing regions of the domain of a function and arelated subdivision method that keeps only those regions where solutionspotentially exist. Compared to the conventional interval analysismethods, the new methods proposed here have a time and memory complexitywhich is the n^(th) root of that of the corresponding conventionalmethod, where n is the “order” of the method used. These methods may beemployed in the context of branch and bound solution search methods thatuse interval analysis described in the previous sections.

[0015] The method uses a Taylor Form inclusion function of the functionf(x) whose solutions are of interest, where x is a vector having anarbitrary number of components. If f(x) can be expanded in the form of apolynomial T(x) with interval coefficients, the method proceeds byseparating the function T(x) into two components: 1.) T_(n)(x), which iscomposed of the terms of T(x) with degree at most n, and can be referredto as the low order component; and 2.) T^(n)(x), which is composed ofthe terms of T(x) with degree greater than n, and can be referred to asthe high order component. Therefore, T(x)=T_(n)(x)+T^(n)(x), where theinteger n is referred to as the “order” of the method.

[0016] The function f(x) on the interval vector domain X can beapproximated by f_(n)(x), which may be expressed in terms of the twocomponents of the function as follows: f_(n)(x)=T_(n)(x)+T^(n)(X) (X isan interval vector containing x). Replacing x by X means we bound thatfunction of variable x on the domain X, i.e. T^(n)(X) is someappropriate inclusion function associated with T^(n)(x). The output ofT^(n)(X) is a one dimensional real interval that represents the errorintroduced by approximating f(x) by f_(n)(x) on the domain X.

[0017] A possible solution region is determined by solving forf_(n)(x)=0, which is now an equation of order n, lower than the order ofthe original equation. If n is low enough, closed form solutions mayexist. Such is the case for univariate equations of order up to 4, aswell as all multivariate equations or first order. Higher orderequations can still be used, but the solution methods will be inexact,i.e. they will return solution regions wider than necessary. Once thesolution sub-region S is determined, a cropping formula is used toidentify one or more sub-regions of the domain X-S which are guaranteednot to contain solutions of the function. The one or more regions may bethen be removed from consideration. Optionally, one or more of theremaining regions of the domain of the function may then be subdividedin preparation for additional iterations of the search.

[0018] First order methods are of particular interest since the croppingformulas are exact and do not introduce any further error. The croppingformulas for the first order method may be expressed as follows:$X_{k}^{*} = {X_{k}\bigcap{{- \left( {c + {\sum\limits_{{i = 0},{i \neq k}}^{n}{a_{i}X_{i}}} + {\Diamond \quad {{NL}(X)}}} \right)}/a_{k}}}$

[0019] where:

[0020] X_(k) ^(*) is the solution-containing portion of the potentialsolution region defined along the k^(th) dimension, i.e. the k^(th)component of the cropped solution containing region X^(*);

[0021] X_(i) is the i^(th) component of the interval vector domain X;

[0022] X_(k) is the k^(th) component of the interval vector domain X

[0023] a_(i) is the Taylor series coefficient for x_(i);

[0024] a_(k) is the Taylor series coefficient for x_(k);$c + {\sum\limits_{{i = 0},{i \neq k}}^{n}{a_{i}X_{i}}}$

[0025] represent the linear (and any constant) terms of the Taylor Forminclusion finction (excepting the term for X_(k)); and

[0026]

NL(X) represent the bounds on X defined by the non-linear terms of theTaylor series expansion.

[0027] In one application, the method identifies one or more portions ofa potential solution-containing region as non-solution-containing. Onceidentified, these one or more portions are removed from the region, andthe remaining portion of the region then subdivided in preparation foradditional iterations of the search.

[0028] The invention also provides a Corner Taylor Form inclusionfunction that can be used with the foregoing methods. This Corner TaylorForm inclusion function is an interval extension of the expansion of thefunction f(x) at the comer of a potential solution-containing regionthat is closest to the origin. This Corner Taylor Form inclusionfunction has the advantage that it always produces results with lessexcess width than the natural inclusion function. This is in contrast tothe Centered Taylor Form inclusion function, which usually has lessexcess width than the natural inclusion function only when the widths ofthe input intervals are smaller than a threshold that depends on thefunction being investigated.

[0029] Other systems, methods, features and advantages of the inventionwill be or will become apparent to one with skill in the art uponexamination of the following figures and detailed description. It isintended that all such additional systems, methods, features andadvantages be included within this description, be within the scope ofthe invention, and be protected by the accompanying claims.

BRIEF DESCRIPTION OF THE DRAWINGS

[0030] The components in the figures are not necessarily to scale,emphasis instead being placed upon illustrating the principles of theinvention. In the figures, like reference numerals designatecorresponding parts throughout the different views.

[0031]FIG. 1 is a flowchart of a conventional branch and bound searchingmethod employing interval analysis.

[0032]FIG. 2 is a flowchart of one embodiment of a method of identifyingone or more regions of the domain of a function that are guaranteed notto contain solutions of the function.

[0033]FIG. 3A illustrates the use of a first order cropping formula toidentify for removal one or more portions of a potentialsolution-containing region.

[0034] FIGS. 3B-3C illustrates the process of subdividing the remainingportion of a potential solution-containing region after application ofthe first order cropping formula of FIG. 3A, followed by additionalapplications of the cropping formula over the subdivided regions.

[0035]FIG. 4 is a flowchart of a searching method in which one or morepotential solution-containing regions are handled in parallel.

[0036]FIG. 5 is a simplified block diagram of system for performing themethod of FIG. 4.

[0037]FIG. 6 is a proof that the order in which each dimension of apotential solution-containing region is cropped does not affect theoutput.

[0038] FIGS. 7A-7B is a proof that the width of the intervals generatedusing the Corner Taylor Form inclusion function is always less than thatgenerated using the natural inclusion function.

[0039] FIGS. 8A-8B illustrate the increased convergence rate andprecision possible using Corner Taylor Form inclusion functions alone.

[0040] FIGS. 8C-8D illustrate the increased convergence rate possibleusing the method of FIG. 2. The numbers on top of each figure denote thenumber of iterations that were required to converge to the solution.

DETAILED DESCRIPTION

[0041] For purposes of this disclosure, the term “solution(s)” will beused to refer collectively to solution(s), root(s), optimizedsolution(s), solution(s) subject to constraints, minimum, minima,maximum, or maxima of a function of one or more variables.

[0042] Also, for purposes of this disclosure, a memory is any processorreadable medium, including without limitation RAM, ROM, EPROM, EEPROM,PROM, disk, hard disk, floppy disk, CD-ROM, DVD, semiconductor memory,etc.

[0043] Moreover, the term “region” refers to any n-dimensional region,where n is an integer of one or more, of the domain of a function,including but not limited to a line segment, box, area, cube, etc.

[0044]FIG. 2 is a flowchart of one embodiment of a method of identifyingone or more regions of the domain of a function where solutions areguaranteed not to exist. This method may be employed in the context ofbranch and bound methods, such as that illustrated in FIG. 1, that useinterval analysis to search for solutions of functions. For example, inone embodiment, the method of FIG. 2 may be employed as part of block110 in FIG. 1 to exclude one or more portions of the potentialsolution-containing region that are guaranteed not to contain solutionsof the function. Then, the remaining portion of the region would besubdivided and placed on the input queue.

[0045] Referring to FIG. 2, in step 202, the method obtains a TaylorForm inclusion function, such as the Corner Form or the Centered Form,of the function f(x), where x is a vector having an arbitrary number ofcomponents.

[0046] The method proceeds to step 204, where the polynomial T(x), withinterval coefficients associated with the Taylor Form inclusion functionchosen, is separated into two components: 1.) T_(n)(x), which iscomposed of the terms of T(x) with degree at most n, and can be referredto as the low order component; and 2.) T^(n)(x), which is composed ofthe terms of T(x) with degree greater than n, and can be referred to asthe high order component. Therefore, T(x)=T_(n)(x)+T^(n)(x), where theinteger n is referred to as the degree of the method.

[0047] In step 208 the function f(x) on the interval vector domain X isbounded by f_(n)(x), which may be expressed in terms of the twocomponents of the function as follows: f_(n)(x)=T_(n)(x)+T^(n)(X) (X isan interval vector containing x). Replacing x by X means we bound thatfunction of variable x on the domain X by computing the associatedinclusion function on X. Then, in this embodiment, the solutions to theoriginal equation f(x)=0 are a subset of the solution set of theequation f_(n)(x)=0, which is now an equation of order n, lower than theorder of the original equation. If n is low enough, closed formsolutions may exist. Such is the case for univariate equations of orderup to 4, as well as all multivariate equations or first order. Higherorder equations can still be used, but the solution methods will beinexact, i.e. they will return solution regions wider than necessary.Even in this case, the solutions of the equation f(x)=0 are still asubset of the solution set of the equation f_(n)(x)=0. Once the possiblesolution sub-region S is determined, a cropping formula derived from oneor both of the components is used to identify one or more sub-regions ofthe domain X-S which are guaranteed not to contain solutions of thefunction. These one or more regions may be then be removed fromconsideration. Optionally, one or more of the remaining regions of thedomain of the function may then be subdivided in preparation foradditional iterations of the search. In one embodiment, the croppingformula may be used to successively identify portions of the regionwhere solutions do not exist dimension by dimension, and it can be shownthat the order in which each dimension of the potentialsolution-containing region is cropped does not affect the output (forproof of this, please see FIG. 6).

[0048] Step 208 is followed by optional step 210 where the one or moreregions are removed from consideration, and one or more of the remainingregions of the domain of the function are subdivided in preparation foradditional iterations of the search.

[0049] First order methods are of particular interest since the croppingformulas are exact and do not introduce any further error. The croppingformulas for the first order method may be expressed as follows:$X_{k}^{*} = {X_{k}\bigcap{{- \left( {c + {\sum\limits_{{i = 0},{i \neq k}}^{n}{a_{i}X_{i}}} + {\Diamond \quad {{NL}(X)}}} \right)}/a_{k}}}$

[0050] where:

[0051] X_(k) ^(*) is the solution-containing portion of the potentialsolution region defined along the k^(th) dimension, i.e. the k^(th)component of the cropped solution containing region X^(*);

[0052] X_(i) is the i^(th) component of the interval vector domain X;

[0053] X_(k) is the k^(th) component of the interval vector domain X

[0054] a_(i) is the Taylor series coefficient for x_(i);

[0055] a_(k) is the Taylor series coefficient for x_(k);$c + {\sum\limits_{{i = 0},{i \neq k}}^{n}{a_{i}X_{i}}}$

[0056] represent the linear (and any constant) terms of the Taylor Forminclusion function (excepting the term for X_(k)); and

[0057]

NL(X) represent the bounds on X defined by the non-linear terms of theTaylor series expansion.

[0058] An example application of this cropping formula is illustrated inFIG. 3A. Referring to FIG. 3A, numeral 300 identifies a potentialsolution region X for the function f(x). For purposes of illustrationonly, and without limitation, the region is shown as having twodimensions, X₁ and X₂, but it should be appreciated that regions arepossible having any number n of dimensions, where n is an integer of oneor more.

[0059] Numeral 306 identifies a portion of the region excluded byapplication of the cropping formula (1) to the first component X₁, andnumeral 304 identifies a portion of the region excluded by applicationof the cropping formula (1) to the second component X₂.

[0060] As can be seen, the remaining region X^(*), identified by 308, isindependent of the order in which the two portions 304 and 306 areexcluded from the region 300.

[0061]FIG. 3B illustrates how the remaining portion 308 of the potentialsolution-containing region may be subdivided into portions 310 a, 310 b,310 c, and 310 d, and FIG. 3D illustrates how the cropping formula maybe applied to each of the subdivided portions to identify additional(cross-hatched) portions that are guaranteed not to contain solutions.These additional crosshatched portions may be removed, the remainingportions subdivided, and the cropping formula applied again. Thisprocess may continue to iterate until the portions that remainadequately define solutions at a suitable level of precision.

[0062] The invention also provides a Corner Taylor Form inclusionfunction that can be used with any of the foregoing methods. Forexample, referring to FIG. 1, this Corner Taylor Form inclusion functionmay be used as the inclusion function in step 106. This Corner TaylorForm inclusion finction is an interval extension of the expansion of thefunction f(x) at the comer closest to the origin of a potentialsolution-containing region.

[0063] This Corner Taylor form inclusion function can be expressed asfollows:${\Diamond \quad {T_{p}\left( \overset{\_}{X} \right)}} = {\sum\limits_{\substack{i \in N^{n} \\ {{\overset{\_}{x}}_{0} = {\inf {({\overset{\_}{X}})}}}}}{\frac{p^{(i)}\left( {\overset{\_}{x}}_{0} \right)}{\left\{ {i!} \right\}*}\left\{ \left( {\overset{\_}{X} - {\overset{\_}{x}}_{0}} \right)^{i} \right\}*}}$

[0064] where:

{overscore (x)}(a real vector)=(x ₀ ^(l) ^(₀) , . . . , x _(n) ^(l)^(_(n)) ), x∈R ^(n+1) , i∈N ^(n+1)

[0065] {overscore (x)}₀ is a real vector defining a corner of thepotential solution region and the origin of the Taylor series expansion;${{\overset{\_}{X}\quad \left( {{an}\quad {interval}\quad {vector}} \right)} = \left( {X_{o}^{i_{0}},\ldots \quad,X_{n}^{i_{n}}} \right)},{X \in {I(R)}^{n + 1}},{{i \in N^{n + 1}};}${x}^(*) = x₀ ⋅ …   ⋅ x_(n);${{p^{(i)}\left( {\overset{\_}{x}}_{0} \right)} = {\frac{\partial^{i_{o} + \ldots \quad + i_{n}}p}{{\partial^{i_{0}}x_{0}}\ldots \quad {\partial^{i_{n}}x_{n}}}\left( {\overset{\_}{x}}_{0} \right)}};{{{and}i} \neq {\left( {{i_{0}!},\ldots \quad,{i_{n}!}} \right).}}$

[0066] This Corner Taylor Form inclusion function has the advantage thatit always produces results with less excess width than the naturalinclusion function (for a proof of this, please see FIGS. 7A-7B). Thisis in contrast to the Centered Taylor Form inclusion function, whichusually has less excess width than the natural inclusion function onlywhen the widths of the input intervals are smaller than a threshold thatdepends on the function being investigated.

[0067] This Corner Taylor Form inclusion function may be stored in amemory and accessed by a processor at an indefinite point of time in thefuture for a variety of uses, e.g., in determining output functionranges, or in the method of FIG. 2. Storage of the Corner Taylor Forminclusion function may also provide efficiencies in the case in whichother potential solution-containing regions share in common the cornerfrom which the inclusion function is derived. When these other regionsare encountered, the stored inclusion function may be retrieved, andused for purposes of defining output function ranges, or excludingportions of the solution-containing region that are guaranteed not tocontain solutions.

[0068] The Corner Form Taylor inclusion ifunction and method of FIG. 2each allow for greatly increased convergence rates, although the impactof the method of FIG. 2 is greater. In particular, the n^(th) orderembodiment of the method of FIG. 2 will reduce the time and memorycomplexity by an order of the n^(th) root of the original complexity ofconventional searches (employing the natural inclusion function andbinary subdivision). For example, if the conventional approach requires10,000 iterations to find potential solution-containing regions, a firstorder embodiment of the method of FIG. 2 will require only 100iterations (square root of 10,000) to find an equivalent set ofpotential solution-containing regions.

[0069] The increased convergence rates possible through Corner FormTaylor inclusion functions and the method of FIG. 2 is illustrated inFIGS. 8A-8D, in which numeral 800 in all four figures identifiessolutions to the function sin(xy)=0 around the point (1,1) (the twoparallel curves), with a solution accuracy of 10⁻⁵. FIG. 8A illustratesthe regions examined in the case of a search method employing thenatural inclusion function and binary subdivision; FIGS. 8B and 8C eachillustrate the regions examined in the case of a search method employingthe Corner Form inclusion function and binary subdivision; and FIG. 8Dillustrates the regions examined in the case of a search methodemploying the Corner Form Taylor inclusion function and the 1^(st) ordermethod of FIG. 2 followed by subdivision. The method of FIG. 8A resultedin the examination of ˜12M solution regions, and ˜45M evaluations; themethod of FIGS. 8B/8C resulted in the examination of ˜1.7M solutionregions, and ˜7M evaluations; and the method of FIG. 8D resulted in theexamination of ˜3142 solution regions, and 7274 evaluations.Consequently, it can be seen that the impact of the Corner Taylor Forminclusion method, although significant, is exceeded by that of themethod of FIG. 2.

[0070] Referring to FIG. 3, a method of searching for solutions of afunction is illustrated which comprises handling p potentialsolution-containing regions in parallel, where p is an integer of one ormore.

[0071] The potential solution regions may be placed on a global inputqueue, and then each separately analyzed for solutions. To begin, asindicated by steps 302(1), 302(2), 302(3), . . . , 302(p), each regionmay first be placed on a separate local input queue. Then, as indicatedby steps 304(1), 304(2), 304(3), . . . , 304(p), each of the regions maybe separately examined in parallel for solutions using, for example, themethod of FIG. 2 for identifying non-solution containing regions of thedomain of a function, with or without the use of the Corner Taylor Forminclusion function.

[0072] Referring to FIG. 5, a system 500 for performing the method ofFIG. 4 includes a plurality p of search engines, identified respectivelywith numerals 508(1), 508(2), . . . , 508(p), where each of the searchengines performs in parallel a search for solutions of a function overone or more different potential solution-containing regions using, forexample, the method of FIG. 2 for identifying non-solution-containingregions of the domain of the function, with or without use of the CornerTaylor Form inclusion function.

[0073] The system also include a controller 502, one or more globalmemories 504, and one or more local memories for each of the searchengines, identified respectively with numerals 508(1), 508(2), . . . ,508(p). As illustrated, the controller 502 has read and write access tothe one or more global memories 504, and also write access to each ofthe local memories for each of the search engines, identifiedrespectively with numerals 506(1), 506(2), . . . , 506(p). Moreover,each search engine has write access to the one or more global memories504, and read and write access to the one or more local memoriesassigned to that search engine.

[0074] The controller 502 maintains the global input queue on the one ormore global memories 504. It addition, it parcels out the potentialsolution regions from the global input queue maintained on the one ormore global memories 504 to each of the p local input queues maintainedon the one or more local memories 506(1), 506(2), . . . , 506(p), asappropriate. Each of the search engines maintains the local input queuecontaining one or more potential storage regions assigned to the searchengine. Each of the search engines also adds to the global solutionqueue maintained on the one or more global memories 504 as appropriate.

[0075] The controller 502 and each of the search engines 508(1), 508(2),. . . , 508(p) may each be implemented as hardware, software, or acombination of hardware and software. Example implementations includewithout limitation one or more synchronous or asynchronous integratedcircuit chips, e.g., ASICS, embodying the foregoing method, or one ormore processors including but not limited to digital signal processors(DSPs) or microprocessors for executing software embodying the foregoingmethod and stored on a memory accessible by the processor.

[0076] Moreover, it should be appreciated that embodiments are possiblein which one or more than one potential solution regions are assigned toeach of the p search engines.

[0077] Each and any of the methods illustrated or described in thisdisclosure may be tangibly embodied as a computer program product or asa series of instructions stored or otherwise embodied on a memory.

[0078] While various embodiments of the invention have been described,it will be apparent to those of ordinary skill in the art that many moreembodiments and implementations are possible that are within the scopeof this invention.

What is claimed is:
 1. A method of identifying one or more regions ofthe domain of a function that do not contain solutions comprising:obtaining a Taylor Form inclusion function; separating the Taylor Forminclusion function into first and second components, the first componentbeing of order n or less, and the second component being of ordergreater than n, where n is an integer; and using a cropping formuladerived from one or both of the components of the Taylor Form inclusionfunction to identify the one or more regions which do not containsolutions.
 2. The method of claim 1 wherein the first componentcomprises the linear and constant terms of the Taylor Form inclusionfunction.
 3. The method of claim 1 wherein the cropping formula issuccessively applied to each of the dimensions of a potentialsolution-containing region.
 4. The method of claim 3 wherein the resultof applying the cropping formula is independent of the order in which itis applied to the dimensions of the potential solution-containingregion.
 5. The method of claim 2 wherein the cropping formula is a firstorder cropping formula, which can be expressed as follows:$X_{k}^{*} = {X_{k}\bigcap{{- \left( {c + {\sum\limits_{{i = 0},{i \neq k}}^{n}{a_{i}X_{i}}} + {\Diamond \quad {{NL}(X)}}} \right)}/a_{k}}}$

where: X_(k) ^(*) is the solution-containing portion of the potentialsolution region defined along the k^(th) dimension, i.e. the k^(th)component of the cropped solution containing region X^(*); X_(i) is thei^(th) component of the interval vector domain X; X_(k) is the k^(th)component of the interval vector domain X a_(i) is the Taylor seriescoefficient for x_(i); a_(k) is the Taylor series coefficient for x_(k);$c + {\sum\limits_{{i = 0},{i \neq k}}^{n}{a_{i}X_{i}}}$

represent the linear (and any constant) terms of the Taylor Forminclusion function (excepting the term for X_(k)); and

NL(X) represent the bounds on X defined by the non-linear terms of theTaylor series expansion.
 6. The method of claim 1 further comprisingexcluding from consideration the one or more regions which have beenidentified as non-solution-containing, and subdividing any remaining oneor more regions of the domain of the function.
 7. The method of claim 6further comprising searching for solutions of the function in thesubdivided portions.
 8. The method of claim 1 wherein, if f(x) can beexpanded in the form of a polynomial T(x) with interval coefficients,the separating step comprises separating T(x) is separated into twocomponents: 1.) T_(n)(x), which is composed of the terms of T(x) withdegree at most n, and can be referred to as the low order component; and2.) T^(n)(x), which is composed of the terms of T(x) with degree greaterthan n, and can be referred to as the high order component.
 9. Themethod of claim 8 wherein the cropping formula is derived according tothe following substeps: approximating the function f(x) on the intervalvector domain X by f_(n)(x), which may be expressed in terms of the twocomponents of the function as follows: f _(n)(x)=T _(n)(x)+T ^(n)(X)where X is an interval vector containing x; and deriving the croppingformula by solving for f_(n)(x)=0, which is now an equation of order n,lower than the order of the original equation.
 10. Any of the methods ofclaims 1, 2, 5, 6, 7, 8, or 9 tangibly embodied as a series ofinstructions stored in a memory.
 11. Any of the methods of claims 1, 2,5, 6, 7, 8, or 9 tangibly embodied as a computer program product.
 12. ACorner Taylor Form inclusion function stored in a memory, wherein theCorner Taylor Form inclusion function is an interval extension of theTaylor expansion of a function f(x) at the corner closest to the originof a potential solution-containing region.
 13. The memory of claim 12,wherein the Corner Taylor Form inclusion function can be expressed asfollows:${\Diamond \quad {T_{p}\left( \overset{\_}{X} \right)}} = {\sum\limits_{\substack{i \in N^{n} \\ {{\overset{\_}{x}}_{0} = {\inf {({\overset{\_}{X}})}}}}}{\frac{p^{(i)}\left( {\overset{\_}{x}}_{0} \right)}{\left\{ {i!} \right\}*}\left\{ \left( {\overset{\_}{X} - {\overset{\_}{x}}_{0}} \right)^{i} \right\}*}}$

where {overscore (x)}(a real vector)=(x ₀ ^(l) ^(₀) , . . . , x _(n)^(l) ^(_(n)) ), x∈R ^(n+1) , i∈N ^(n+1) {overscore (x)}₀ is a realvector defining a corner of the potential solution region and the originof the Taylor series expansion;${{\overset{\_}{X}\quad \left( {{an}\quad {interval}\quad {vector}} \right)} = \left( {X_{o}^{i_{0}},\ldots \quad,X_{n}^{i_{n}}} \right)},{X \in {I(R)}^{n + 1}},{{i \in N^{n + 1}};}${x}^(*) = x₀ ⋅ …   ⋅ x_(n);${{p^{(i)}\left( {\overset{\_}{x}}_{0} \right)} = {\frac{\partial^{i_{o} + \ldots \quad + i_{n}}p}{{\partial^{i_{0}}x_{0}}\ldots \quad {\partial^{i_{n}}x_{n}}}\left( {\overset{\_}{x}}_{0} \right)}};{{{and}i} \neq {\left( {{i_{0}!},\ldots \quad,{i_{n}!}} \right).}}$


14. A method of determining the output range of a function comprising:obtaining a Corner Taylor Form inclusion function which is an intervalextension of the Taylor expansion of a function f(x) at the cornerclosest to the origin of a potential solution-containing region; andusing the Corner Taylor Form inclusion function to determine the outputrange of the function.
 15. The method of claim 14 wherein the CornerTaylor Form inclusion function can be expressed as follows:${\Diamond \quad {T_{p}\left( \overset{\_}{X} \right)}} = {\sum\limits_{{i \in N^{''}}{{{\overset{\_}{x}}_{0} = {\inf {({\overset{\_}{X}})}}}}}{\frac{p^{(i)}\left( {\overset{\_}{x}}_{0} \right)}{\left\{ {i!} \right\}*}\left\{ \left( {\overset{\_}{X} - {\overset{\_}{x}}_{0}} \right)^{i} \right\}*}}$

where {overscore (x)}(a real vector)=(x ₀ ^(l) ^(₀) , . . . , x _(n)^(i) ^(_(n)) ), x∈N ^(n+1) , i∈N ^(n+1) {overscore (x)}₀ is a realvector defining a corner of the potential solution region and the orginof the Taylor series expansion;${{\overset{\_}{X}\left( {{an}\quad {interval}\quad {vector}} \right)} = \left( {X_{0}^{i_{0}},\ldots \quad,X_{n}^{i_{n}}} \right)},{X \in {I(R)}^{n + 1}},{{i \in N^{n + 1}};}${x}^(*) = x₀ ⋅ … ⋅ x_(n);${{p^{(i)}\left( {\overset{\_}{x}}_{0} \right)} = {\frac{\partial^{i_{0} + \ldots + i_{n}}p}{{\partial^{i_{0}}x_{0}}\ldots {\partial^{i_{n}}x_{n}}}\left( {\overset{\_}{x}}_{0} \right)}};{and}$i! = (i₀!, …  , i_(n)!).


16. The method of claim 15 further comprising using the determinedoutput range to identify whether a corresponding region of the domain ofthe function is a potential solution-containing region.
 17. The methodof claim 15 further comprising using the method of claim 1 to identifyone or more portions of the potential solution-containing region whichdo not contain solutions.
 18. The method of claim 17 further comprisingsubdividing any portion of the potential solution-containing regionremaining after exclusion of the one or more portions which have beenidentified as not containing solutions.
 19. The methods of any of claims14, 15, 16, 17, or 18 tangibly embodied as a series of instructionsstored in a memory.
 20. The methods of any of claims 14, 15, 16, 17, or18 tangibly embodied as a computer program product.
 21. A method forsearching for solutions of a function comprising: obtaining p potentialsolution-containing regions, where p is an integer of two or more; andusing the method of claim 1 to perform in parallel p searches, each overa different one of the p potential solution-containing regions.
 22. Asystem for searching for one or more solutions of a function comprising:a plurality p of search engines, where p is an integer of two or more,each configured to use the method of claim 1 to perform in parallel asearch over a different one of p potential solution-containing regions.23. The system of claim 22 further comprising a controller for assigningthe potential solution-containing regions to each the p search engines.24. The system of claim 23 further comprising one or more globalmemories accessible by the controller for maintaining potentialsolution-containing regions for assignment.
 25. A method of identifyingone or more regions of the domain of a function that do not containsolutions comprising: a step for obtaining a Taylor Form inclusionfunction; a step for separating the Taylor Form inclusion function intofirst and second components, the first component being of order n orless, and the second component being of order greater than n, where n isan integer; and a step for using a cropping formula derived from one orboth of the components of the Taylor Form inclusion function to identifythe one or more regions which do not contain solutions.
 26. A method ofdetermining the output range of a function comprising: a step forobtaining a Corner Taylor Form inclusion function which is an intervalextension of the Taylor expansion of a function f(x) at the cornerclosest to the origin of a potential solution-containing region; and astep for using the Corner Taylor Form inclusion function to determinethe output range of the function.
 27. Any of the methods of claims 25 or26 tangibly embodied as a series of instructions stored in a memory. 28.Any of the methods of claims 25 or 26 tangibly embodied as a computerprogram product.
 29. A system for searching for one or more solutions ofa function comprising: the memory of claim 19; and a processorconfigured to access the memory and execute the instructions storedthereon to search for the one or more solutions.